By Chee-Keng Yap

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S. Householder. Principles of Numerical Analysis. McGraw-Hill, New York, 1953. [86] L. K. Hua. Introduction to Number Theory. Springer-Verlag, Berlin, 1982. ¨ [87] A. Hurwitz. Uber die Tr¨ agheitsformem eines algebraischen Moduls. Ann. Mat. , 3(20):113–151, 1913. [88] D. T. Huynh. A superexponential lower bound for Gr¨ obner bases and Church-Rosser commutative Thue systems. Info. and Computation, 68:196–206, 1986. [89] C. S. Iliopoulous. Worst-case complexity bounds on algorithms for computing the canonical structure of finite Abelian groups and Hermite and Smith normal form of an integer matrix.

217] C. K. Yap. A new lower bound construction for commutative Thue systems with applications. J. of Symbolic Computation, 12:1–28, 1991. [218] C. K. Yap. Fast unimodular reductions: planar integer lattices. IEEE Foundations of Computer Science, 33:437–446, 1992. [219] C. K. Yap. A double exponential lower bound for degree-compatible Gr¨ obner bases. Technical Report B-88-07, Fachbereich Mathematik, Institut f¨ ur Informatik, Freie Universit¨ at Berlin, October 1988. [220] K. Yokoyama, M. Noro, and T.

Then m, n ∈ Z has the property that m | n iﬀ (m) ⊇ (n), “agreeing” with our definition. In general, the relationship between ideal quotient and divisor property is only uni-directional: for ideals I, J ⊆ D, we have that I ⊇ IJ and so I divides IJ. The GCD of a set S of ideals is by definition the smallest ideal that divides each I ∈ S, and we easily verify that GCD(S) = I. I∈S For I = (a1 , . . , am ) and J = (b1 , . . , bn ), we have GCD(I, J) = I + J = (a1 , . . , am , b1 , . . , bn ). (1) So the GCD problem for ideals is trivial unless we require some other conditions on the ideal generators.